Optimal Design of a High-Speed Single-Phase Flux

energies Article

Optimal Design of a High-Speed Single-Phase Flux Reversal Motor for Vacuum Cleaners Vladimir Dmitrievskii 1,2 , Vladimir Prakht 1,2 , Vadim Kazakbaev 1,2, * 1

2

*

and Sergey Sarapulov 1

Department of Electrical Engineering and Electric Technology Systems, Ural Federal University, Yekaterinburg 620002, Russia; [email protected] (V.D.); [email protected] (V.P.); [email protected] (S.S.) EMACH LLC, Yekaterinburg 620002, Russia Correspondence: [email protected]; Tel.: +7-343-375-4507

Received: 3 October 2018; Accepted: 26 November 2018; Published: 29 November 2018







Abstract: This paper describes the design of a single-phase high-speed flux reversal motor (FRM) for use in a domestic application (vacuum cleaner). This machine has a simple and reliable rotor structure, which is a significant advantage for high-speed applications. An FRM design in which the inner stator surface is entirely used allows it to decrease its volume and increase its efficiency. The mathematical modeling, based on the finite element method, and the optimal design of the high-speed single-phase FRM are described. The criterion of optimization and the selection of a proper optimization algorithm are discussed. Since the finite element method introduces a small but quasi-random error due to round-off accumulation and choosing the mesh, etc., the Nelder-Mead method, not requiring the derivatives calculation, was chosen for the optimization. The target parameter of the optimization is built for the motor efficiency when operating at different loads. Calculations show that the presented approach provides increasing motor efficiency during the optimization, particularly at underload. Keywords: electrical machine modeling and optimal design; flux reversal motor; high-speed motor; permanent magnet machines; permanent magnet machine drives; special machines; vacuum cleaner

1. Introduction An important characteristic of vacuum cleaners (Figure 1a) and air blowers (Figure 1b) is the suction power, which directly depends on the power of the applied electric motor [1]. One prospective of such devices is the increase in suction power without increasing the size and mass of devices. In addition, to improve reliability and durability, it is necessary to eliminate the collector-and-brush motor assembly unit. However, these requirements cannot be satisfied with the use of an AC commutator (brushed, universal) motor (Figure 2a), traditionally used in vacuum cleaners and air blowers [2,3]. Therefore, for the application under consideration, the use of brushless electric motors is promising [2–6]. A high-speed three-phase brushless synchronous motor for vacuum cleaners and air blowers [2] is shown in Figure 2b and a two-phase switched reluctance motor [3] is demonstrated in Figure 2c. In high-speed vacuum cleaners and air blowers the use a type of motor with magnets on the stator is expected, which is known as a flux reversal motor [7,8]. Two possible designs of the high-speed flux reversal motor are shown in Figure 3.

Energies 2018, 11, 3334; doi:10.3390/en11123334

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(a) (a)

(b) (b)

Figure Considered motor applications: (a) vacuum cleaner [9] (b) air blower [10] Figure1.1.1.Considered Consideredmotor motorapplications: applications:(a) (a)vacuum vacuumcleaner cleaner[9] [9]; (b)air airblower blower[10] [10]... Figure ;;(b)

(a) (a)

(b) (b)

(c) (c)

Figure2.2.Design Designof ofhigh-speed high-speedmotors: motors:(a) (a)single-phase single-phaseAC ACcommutator commutatormotor motor[11] [11]; three-phase Figure Design of high-speed motors: (a) single-phase AC commutator motor [11] (b)three-phase three-phase Figure ;;(b) motor having permanent magnets on the rotor [2]; (c) two-phase switched reluctance motor [3]. motorhaving havingpermanent magnets on the rotor [2] [2];;(c) (c)two-phase two-phaseswitched switchedreluctance reluctancemotor motor[3]. [3]. motor

A flux reversal many advantages in comparison withwith otherother typestypes of high-speed A flux reversal reversalmachine machine(FRM) (FRM)has has many advantages in comparison comparison of highhighA flux machine (FRM) has many advantages in with other types of machines [7]. speed machines [7]. speed machines [7]. When compared commutator motors [11],[11], the main advantages of an FRM are FRM the following: When comparedto toAC AC commutator motors the main main advantages advantages of an an are the the When compared to AC commutator motors [11], the of FRM are following: • The FRM is more reliable and its lifetime is longer, as it has no collector assembly and no brushes; following: • The FRM heat removal from the rotor is not required rotor does not contain any heating The morereliable reliable and itslifetime lifetime longer,because asitithas hasthe nocollector collector assembly andno nobrushes; brushes; •• The FRM isismore and its isislonger, as no assembly and elements such as coils a collector. This also increases of not the contain rotor bearings; The heatremoval removal fromor the rotorisisnot not required becausethe thereliability rotordoes does anyheating heating •• The heat from the rotor required because the rotor not contain any • elements The toothed rotor production is very simple and not necessarily required. Its only active part is suchas ascoils coilsor oraacollector. collector.This Thisalso alsoincreases increasesthe thereliability reliabilityof ofthe therotor rotorbearings; bearings; elements such a magnetic made of laminated The toothedcore rotor production verysteel; simpleand andnot notnecessarily necessarilyrequired. required.Its Itsonly onlyactive activepart partisis •• The toothed rotor production isisvery simple • aaThe FRM mass is much less than that of the AC commutator motor with brushes; magneticcore coremade madeof oflaminated laminatedsteel; steel; magnetic • The FRM mass is much less than that ofthe the AC commutator motorwith with brushes; • efficiency of the FRM is higher than thatAC of commutator the AC commutator motor with brushes. • The FRM mass is much less than that of motor brushes; • The efficiency of the FRM is higher than that of the AC commutator motor withbrushes. brushes. • The of the FRM is higher thanthree-phase that of the and AC commutator with Theefficiency main advantages of the FRM over single-phase motor synchronous motors having The main advantages of the FRM over three-phase and single-phase synchronous motors having permanent magnets (PM) on the rotor [2,12,13] are the following: The main advantages of the FRM over three-phase and single-phase synchronous motors having permanentmagnets magnets(PM) (PM)on on therotor rotor[2,12,13] [2,12,13]are arethe thefollowing: following: permanent • The toothed rotor of thethe FRM is simpler and more reliable, without permanent magnets; • The toothed rotor of the FRM is simpler and more reliable,without withoutpermanent permanentmagnets; magnets; •• The toothed rotor of the FRM is simpler and more reliable, High-speed synchronous motors with magnets on the rotor often have an additional retaining • High-speed synchronous motors with magnets on the rotor often have an additional retaining • High-speed synchronous with magnets on the rotor often have an additional retaining ring on the rotor to fix themotors magnets; ringon onthe therotor rotorto tofix fixthe themagnets; magnets; • ring The FRM FRM rotor is is simpler, simpler, more reliable, and does not require balancing. • The rotor more reliable,and anddoes doesnot notrequire requirebalancing. balancing. • The FRM rotor is simpler, more reliable, In addition addition to to this, this, the the FRM FRM single-phase single-phase inverter inverter is simpler simpler and and cheaper cheaper than than a three-phase three-phase In In addition to this, the FRM single-phase inverter isis simpler and cheaper than aa three-phase synchronous motor with PMs on the rotor. synchronousmotor motorwith withPMs PMson onthe therotor. rotor. synchronous The FRM has similar advantages at high-speed high-speed applications applications to to the the Switched Switched Reluctance Reluctance Motor Motor The FRM FRM has has similar similar advantages advantages at at The high-speed applications to the Switched Reluctance Motor (SRM), since since both both have have aa simple simple and and reliable reliable rotor. (SRM), rotor. (SRM), since compared both have atosimple and reliable rotor. When the two-phase SRM [14], additional advantages of the are When compared to the two-phase SRM [14], thethe additional advantages of the the FRMFRM are the the When compared to the two-phase SRM [14], the additional advantages of FRM are the following: following: following: • Only Only half half aa period period is is used used to to generate generate torque torque in in the the SRM, SRM, while while the the FRM FRM can can do do it it for for almost almost an an •• Only half a period is used to generate torque in the SRM, while the FRM can do it for almost an entire period, period, which which increases increases the efficiency as well as the specific torque and power; entire theefficiency efficiencyas aswell wellas asthe thespecific specifictorque torqueand andpower; power; entire period, which increases the

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••

The The SRM SRM has has open open stator stator slots slots [14] [14] and and only only about about half half the the stator stator surface surface is is used, used, which which decreases decreases the efficiency as well as the specific torque and power; the efficiency as well as the specific torque and power; • Almost half of the electrical energy received by the SRM returns to its inverter and does not • Almost half of the electrical energy received by the SRM returns to its inverter and does not transform to mechanical energy, which increases the winding current as well as increasing the transform to mechanical energy, which increases the winding current as well as increasing the inverter cost and size; inverter cost and size; • The single-phase inverter (control circuit of the motor) for the FRM is simpler and cheaper • The single-phase inverter (control circuit of the motor) for the FRM is simpler and cheaper compared to the inverter for the two-phase SRM. compared to the inverter for the two-phase SRM. Given the advantages noted above in high-speed applications where a large starting torque is Given the advantages noted above in high-speed applications where a large starting torque is not required (vacuum cleaners, hand dryers, blowers, etc.), the use of a single-phase FRM is a good not required (vacuum cleaners, hand dryers, blowers, etc.), the use of a single-phase FRM is a good alternative to the SRM and synchronous motors with PMs on the rotor. alternative to the SRM and synchronous motors with PMs on the rotor. Recently, a large number of three-phase configurations of the FRM were described in the Recently, a large number of three-phase configurations of the FRM were described in the literature [15–17]. However, the three-phase FRMs require both a more expensive power converter literature [15–17]. However, the three-phase FRMs require both a more expensive power converter and a complicated control system. Therefore, in the considered application, the use of a single-phase and a complicated control system. Therefore, in the considered application, the use of a single-phase FRM that has a simple converter design [7] is more preferable. FRM that has a simple converter design [7] is more preferable. The design of a single-phase FRM is not so often discussed in the literature. At present, only The design of a single-phase FRM is not so often discussed in the literature. At present, several various configurations of a single-phase FRM are known. Ref. [18] describes the first known only several various configurations of a single-phase FRM are known. Ref. [18] describes the first configuration of the single-phase FRM. This machine has two stator teeth with two magnetic poles known configuration of the single-phase FRM. This machine has two stator teeth with two magnetic on each, and three rotor teeth (Figure 3a). A similar high-speed FRM for vacuum cleaners is described poles on each, and three rotor teeth (Figure 3a). A similar high-speed FRM for vacuum cleaners is in Reference [19]. described in Reference [19].

(a)

(b)

Single-phase flux reversal machine (FRM) designs: designs: (a) (a) the the design design of of an FRM according to Figure 3. Single-phase Reference [18]; (b) the design of an FRM according to References [7,20]. References [7,20].

Single-phase FRMs, FRMs,inin which surface is entirely are described in Single-phase which the the innerinner statorstator surface is entirely used, areused, described in Reference Reference [7,20]. This FRM operates at a very high speed ( ≈ 28,000 rpm). This machine was considered [7,20]. This FRM operates at a very high speed (≈28,000 rpm). This machine was considered for for driving an angular grinder (electric power hand tool). Figure 3b shows design of that FRM. driving an angular grinder (electric power hand tool). Figure 3b shows the the design of that FRM. Its Its stator consists of a stator core with four teeth and a four-pole single-phase winding. On the internal stator consists of a stator core with four teeth and a four-pole single-phase winding. On the internal surface of of each each stator stator tooth tooth there there are are two two fixed fixed PMs. PMs. The The direction direction of of the the magnetization magnetization of of the the nearest nearest surface PMs fixed fixed on the adjacent adjacent teeth teeth is same. In In addition addition to to this, this, the the new new FRM’s FRM’s design design has has aa rotor rotor with with PMs on the is the the same. four teeth. four teeth. When compared compared to to the modifications of of FRMs FRMs [18,19], [18,19], the the advantages of the the When the well-known well-known modifications advantages of considered design (Figure 3b) are the following: (1) increasing mechanical power density and efficiency considered design (Figure 3b) are the following: (1) increasing mechanical power density and of the motor, to betterdue utilization of utilization the stator internal surface; (2) no inherent efficiency of due the motor, to better of the stator internal surface; uncompensated (2) no inherent radial forces; and (3) an increase in the lifetime of the bearings as a result of the symmetry of the of motor uncompensated radial forces; and (3) an increase in the lifetime of the bearings as a result the with an even number of rotor teeth. symmetry of the motor with an even number of rotor teeth. Various approaches approaches totothe themodeling modelingof of FRM were presented in References [7,18,19]. Various thethe FRM were presented in References [7,18,19]. The The optimization the geometry of thewith FRM with three rotor teeth3a) (Figure 3a) was discussed in optimization of theofgeometry of the FRM three rotor teeth (Figure was discussed in Reference [18]. However, the criterion of optimization was not presented. The calculation of the performances

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Reference [18]. However, the criterion of optimization was not presented. The calculation of the performances of the FRM was carried out using parameters such as the “fringing coefficient”, obtained from an FEM (finite element method) model. Furthermore, it was not mentioned in the paper how the steel saturation was accounted for in the model. Another approach was presented in Reference [19]. The Simulink implemented model of an FRM is based on data on inductance, and back EMF (electromotive force) obtained from an FEM model. The model does not take into consideration losses in the magnetic cores and magnets, which may be essential at high frequencies. When designing the FRM for a variable-speed application, it is necessary to take into account the lack of FRM inherent in all motors with magnets. Their magnetic field cannot be switched off or weakened at low loads. Therefore, with a suboptimal motor design for the rated mode, the losses at underload will be significant, and the efficiency will be low. This necessitates the minimization of FRM losses over a wide range of load modes. In this paper, the optimal design technique of a high-speed FRM for a vacuum cleaner, choosing the optimization criterion, and the selection of the optimization algorithm are described. The characteristics of single-phase FRMs designed without the optimization and those of single-phase FRMs designed with the optimization are compared. An FEM-based mathematical model was developed for this optimization. The case of feeding the FRM with the voltage of a rectangular waveform is considered. To reduce the number of boundary problems required, the motor symmetry is studied. The iterating procedure followed to converge the model with the mode at a given power is described. 2. Brief Description of the New FRM Mathematical Model A new mathematical model was developed for an optimal FRM design. The rectangular voltage form is assumed in the model. The winding, magnets, and core losses are neglected in solving the boundary problems and are estimated in the post-processing. The flux waveform is trapezoidal and can be obtain by integrating the voltage waveform. The rotor speed is assumed to be a constant during the machine’s electrical cycle (rotating the rotor by 90 mechanical degrees). The integration constant is chosen so that the average current is equal to zero. Thus, only a few combinations of the flux and the rotor position are realized in the machine cycle, and only the boundary problems corresponding to these combinations should be solved. Therefore, considering all possible combinations is not required. A 2D magnetic vector potential approach was applied to solve these boundary problems. This approach involved dividing the calculated area into two subareas corresponding to the rotor and the stator by boundary III (Figure 4), using a joining boundary condition depending on whether the rotor position allowed, using one and the same geometry for all the boundary problems, and considering the rotor and the stator in their own reference frames. Such an approach simplifies the calculation of the total time derivative of the magnetic flux density necessary for the calculation of the core and magnet losses. The geometry of the FRM is symmetrical with respect to its rotation by 90 mechanical degrees and changing the current and magnetization directions. Therefore, it is sufficient to consider only a 14 section of the geometry. The boundaries I and II are linked by an antiperiodic boundary condition where the vector magnetic potential changes its sign. Usually, for the boundary problem to be solved, the current density should be given. However, when the flux Φ is given, the current density J is assumed to be proportional to the unknown winding current I: J = ΞI, (1) where Ξ is the current density at the winding current of 1A. The winding current I should be considered an additional degree of freedom. The corresponding equation is: Φ = 4L

x

AΞdxdy,

(2)

Thus, only a few combinations of the flux and the rotor position are realized in the machine cycle, and only the boundary problems corresponding to these combinations should be solved. Therefore, considering all possible combinations is not required. A 2D magnetic vector potential approach was applied to solve these boundary problems. This approach involved dividing the calculated area into Energies 2018, 11, corresponding 3334 5 of 13 two subareas to the rotor and the stator by boundary III (Figure 4), using a joining boundary condition depending on whether the rotor position allowed, using one and the same geometry for all the boundary problems, and considering the rotor and the stator in their own where is the of magnetic potential, L is the stack length, and 4 is the multiplier, taking Energies A 2018, 11, xz-component FOR PEER REVIEW 5 of 14 reference frames. Such an approach simplifies the calculation of the total time derivative of the into account the reduction of the calculating area. magnetic flux density the calculation the core magnet losses. The geometry of necessary the FRM isforsymmetrical withofrespect toand its rotation by 90 mechanical degrees and changing the current and magnetization directions. Therefore, it is sufficient to consider only a ¼ section of the geometry. The boundaries I and II are linked by an antiperiodic boundary condition where the vector magnetic potential changes its sign. Usually, for the boundary problem to be solved, the current density should be given. However, when the flux  is given, the current density J is assumed to be proportional to the unknown winding current I:

J = I ,

(1)

where  is the current density at the winding current of 1A. The winding current I should be considered an additional degree of freedom. The corresponding equation is:

 = 4L  Adxdy ,

(2)

where A is the z-component of magnetic potential, L is the stack length, and 4 is the multiplier, taking Figure 4. 4. Calculating Calculating area. area. Figure into account the reduction of the calculating area. As a result of solving the set of boundary problems, the current waveform can be found. Then As a result of solving the set of boundary problems, the current waveform can be found. Then the the torque and the winding losses can be estimated in postprocessing. In particular, the torque T can torque and the winding losses can be estimated in postprocessing. In particular, the torque T can be be found as follows: found as follows: x xy( B2 − Bx2 ) +2 ( x2 2− y2 ) Bx By xy ( B y2 −y Bx2 ) + (p x − y ) Bx B y T = 4L dxdy, (3) T = 4 L  dxdy , δair µ0 x2 + y2 (3) air air 0 x 2 + y 2 air gap gap subdomain subdomain

where δairairis is thethe airair gap thickness. gap thickness. It is supposed that the middlesofofthe theimpulses impulsesofofthe the rectangular supply voltage correspond It is supposed that the middles rectangular supply voltage correspond to to the moments of time when the rotor teeth are exactly over the magnetic poles of the stator teeth. the moments of time when the rotor teeth are exactly over the magnetic poles of the stator teeth. Therefore, mode is is to that is is the the part part of of Therefore, the the only only way way to to affect affect the the operating operating mode to choose choose the the duty-cycle duty-cycle that time when the applied voltage is not zero. The duty cycle through the applied voltage determines the time when the applied voltage is not zero. The duty cycle through the applied voltage determines the flux waveform in Equation and flux waveform involved involved in Equation (2). (2). To To calculate calculate the the motor motor mode mode with with the the given given torque torque TTref ref and speed, an algorithm was developed and its flowchart is shown in Figure 5. When the torque is equal to speed, an algorithm was developed and its flowchart is shown in Figure 5. When the torque is equal T the required accuracy, the calculations stop. torefTwith ref with the required accuracy, the calculations stop.

Figure 5. 5. The The FRM FRM modeling modeling flowchart. flowchart. Figure

3. The FRM Optimization Criteria and Procedure Since the vacuum cleaner works in a wide range of speeds and torques depending on the operating mode, the rated efficiency cannot serve as a correct indicator of the energy efficiency of the

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3. The FRM Optimization Criteria and Procedure vacuum cleaner. As a quantitative measure of energy efficiency for a variable-load motor, the average Since the vacuum cleaner works in a wide range of speeds and torques depending on the operating efficiency can be used: mode, the rated efficiency cannot serve as a correct indicator of the energy efficiency of the vacuum cleaner. As a quantitative measure of energy efficiency for a variable-load motor, the average efficiency Pmech i ti av  , (4) can be used: Pai ti ∑ Pmechi ti ηav = , (4) ∑ Pai ti where Pmechi is the mechanical power in different modes; Pai denotes the value of active where Pmechi is the mechanical power in different modes; Pai denotes the value of active electrical electrical power in various modes; andand the proportion of the operating time of a certain mode. ti tis power in various modes; i is the proportion of the operating time of a certain mode. modes of theofoperation of the variable-speed motor by are analogy chosen by analogy with the The modes The of the operation the variable-speed motor are chosen with the recommendations from [21](quadratic for fan load (quadratic dependence of torque recommendations from Reference [21]Reference for fan load dependence of torque and speed) and and speed) and are are given in Table 1. given in Table 1.

 

Load modes included in the criterion. optimization criterion. Table 1. LoadTable modes1.included in the optimization

Mode Mode № Shaft power, ratedvalue value Shaft power,% % of of the the rated Rotational the rated ratedvalue value Rotationalspeed, speed, % % of of the Torque, % of the rated value Torque, % of the rated value Weight coefficient Weight coefficient

1 12.5 12.5 50 50 25 25 1 1

2 37.5 75 50 1

32 100 37.5 100 75 100 50 1 1

3 100 100 100 1

Typically, fan units are operated with a substantial underload for a longer time. Therefore, in fan tounits arearithmetic operatedaverage with efficiency a substantial for2, 3a aslonger time. Reference [21] it Typically, was proposed use the in theunderload modes i = 1, Referenceof[21] was proposed to usecorresponds the arithmetic in the modes i = 1, a measure ofTherefore, the energyinefficiency theitpump motor, which to average ti ~ 1/Paiefficiency . 2, 3 as a measure of the energy efficiency of the pump motor, which corresponds to ti ~ 1/Pai . The vacuum cleaner is designed for cleaning dust from different surfaces with a hard-to-predict The vacuum cleaner is designed for cleaning dust from different surfaces with a hard-to-predict aerodynamic resistance. Therefore, it was accepted that all modes are implemented at the same time. aerodynamic resistance. Therefore, was accepted that all modes are implemented at the same time. Therefore, Equation (4) transforms into Equationit (5): Therefore, Equation (4) transforms into Equation (5):

av  

Pmechi

ηav =  Pa i

∑ Pmechi ∑ Pai

(5)

(5)

This value was chosen thechosen optimization criterion. When the optimization criterion hascriterion many has many This value as was as the optimization criterion. When the optimization local extrema, the heuristic Monte Carlo methods such as genetic algorithms and simulated annealing local extrema, the heuristic Monte Carlo methods such as genetic algorithms and simulated annealing algorithms [22] are usually algorithms [22]applied. are usually applied. The optimization criteria usedcriteria in the used optimal design of electric havemotors fewer extrema. The optimization in the optimal designmotors of electric have fewer extrema. However, using gradient descent optimization algorithms such as the Newton algorithm trust However, using gradient descent optimization algorithms such as the Newtonoralgorithm or trust region methods together with FEM-based models is almost impossible, region[23,24] methods [23,24] together with FEM-based models is almost impossible,because because the the calculations calculations using using FEM FEM are are accompanied accompanied by by the the unpredictable unpredictable pseudorandom pseudorandom computational computational error error [25] and the [25] and the derivatives derivatives calculations calculations of of the the optimization optimization criterion criterion become become complicated. complicated. To overcomeTothese difficulties, type of gradient-free method known as the Nelder–Mead overcome these the difficulties, the type of gradient-free method known as the Nelder–Mead method [25] method is used in this work. This method is based upon the comparison of the criterion [25] is used in this work. This method is based upon the comparison of the values criterion values for for various cases, but not upon their numerical values.values. Therefore, it can be used noisy or nonvarious cases, but not upon their numerical Therefore, it can be for used for noisy or non-smooth smooth functions. Moreover, it can be more effective than some other gradient-free methods, functions. Moreover, it can be more effective than some other gradient-free methods,inin particular, particular, the thePowell Powellmethod method[26]. [26]. The proposed criterion (5)criterion requires(5) therequires calculations of three motor modes one per calculation The proposed the calculations of three motorper modes one calculation step. step. However, as a result, higher efficiency at a wide range of operating modes and not only the at the rated However, as a result, higher efficiency at a wide range of operating modes and notatonly rated conditions can be achieved. It also provides a longer running time for a vacuum cleaner battery conditions can be achieved. It also provides a longer running time for a vacuum cleaner battery with with the FRM assuming the same as the battery. the FRM assuming thecapacity same capacity as the battery. The fixed parameters of the designed FRM for the vacuum are ascleaner follows: speed The fixed parameters of the designed FRM for cleaner the vacuum arethe as rated follows: the rated speed is 28,000 rpm; rated power 1.5 kW; the is outer diameter of thediameter stator is of 54 the mm;stator the length of thethe length of is the 28,000 rpm; the is rated power 1.5 kW; the outer is 54 mm; stator lamination is 40 mm; the air gap is 0.4 mm. The FRM requires magnets with a residual flux density of 1.3 T. Each magnet is divided into eight electrically insulated pieces in the azimuthal direction to reduce the eddy current losses. The magnet thickness is assumed to be 1.7 mm because

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the stator lamination is 40 mm; the air gap is 0.4 mm. The FRM requires magnets with a residual flux density of 1.3 T. Each magnet is divided into eight electrically insulated pieces in the azimuthal direction to reduce the eddy current losses. The magnet thickness is assumed to be 1.7 mm because Energies 2018, 11, x FOR PEER REVIEW 7 of 14 thinner magnets have a higher price per kg and thus would not reduce the motor cost. M330-50A steel grade was chosen. thinner magnets have a higher price per kg and thus would not reduce the motor cost. M330-50A Figure 6 shows steel grade was chosen.the optimization parameters: 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6.

Figure 6 shows the optimization parameters: Outer rotor radius Rr ; Slotrotor bottom radius Outer radius Rr; Rb ; Slot width W; Slot bottom radius Rb; SlotRotor width W; width Zr ; tooth Rotor tooth width Zr; rb ; Rotor bottom radius Rotor bottom radius Stator winding coilrb;number N. Stator winding coil number N.

Figure 6. Calculating area. Figure 6. Calculating area.

4. The Results of the Optimization 4. The Results of the Optimization To obtain thex FOR maximum value of Equation (5), the human-made design (shown in Figure 7a) was Energies 2018, 11, PEER REVIEW To obtain the maximum value of Equation (5), the human-made design (shown in Figure 7a)6 of 14 used as a starting point. Figure 7b shows the optimized design. The optimization parameters before was used as a starting point. Figure 7b shows the optimized design. The optimization parameters cleaner. As a quantitative andvacuum after the optimization are shownmeasure in Table of 2. energy efficiency for a variable-load motor, the average before and after the optimization are shown in Table 2. efficiency can be used: Table 2. FRM parameters (before and after the optimization). Parameter Outer rotor radius Rr , mm

av   Before

Pmechi ti

Pa t  12.9

i i

,

After

(4)

11.7

Slot ibottom Rb , mm power in different 23.5 23 is theradius mechanical modes; Pai denotes the value of active Pmech Slotinwidth W, degrees 45 54.8time of a certain mode. electrical power various modes; and ti is the proportion of the operating

where

Rotor width Zr of the variable-speed 7 6 by analogy with the The modes of tooth the operation motor are chosen Rotor bottom rb , mm recommendations from radius Reference [21] for fan load 8.7 (quadratic dependence 7.2 of torque and speed) and are given in Table 1. Stator winding coil number N 50 64

Table 1. Load modes included in the optimization criterion.

Figures 8–10 show the transient values of the current, the voltage, and the torque depending on Mode № 3, 28,000 rpm, 1.5 kW). 1 2 3 the rotor position for the rated mode (mode Shaft power, % of the rated value 12.5 37.5 100 Rotational speed, % of the rated value 50 75 100 (a) (b) Torque, % of the rated value 25 50 100 Weight coefficient 1 optimization; 1 1 (b) after Figure 7. Simultaneous flux density field (T) in the FRM: (a) before optimization.

Figure 6. Calculating area.

4. The Results of the Optimization To obtain the maximum value of Equation (5), the human-made design (shown in Figure 7a) 8 of 13 was used as a starting point. Figure 7b shows the optimized design. The optimization parameters before and after the optimization are shown in Table 2.

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Parameter Outer rotor radius Rr, mm Slot bottom radius Rb, mm Parameter Slot width degrees Outer rotor W, radius Rr, mm Slot bottom RbZ , mm Rotor toothradius width r Slot width W, degrees Rotor bottom radius rb, mm Rotor tooth width Zr N Stator winding coil number

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Before 12.9 23.5 Before 45 12.9 23.57 458.7 7 50

After 11.7 After23 11.754.8 23 6 54.87.2 6 64

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Rotor bottom radius rb, mm 8.7 7.2 Stator winding coil number N 50 64 torque depending on (a)transient values of the current, the voltage,(b) Figures 8–10 show the and the

the rotor position for the rated mode (mode №(T) 3, in 28,000 rpm, 1.5optimization; kW). optimization; Figure 7. Simultaneous flux density fieldin the(a) FRM: (a) before after Figure 7. Simultaneous flux density field (T) the FRM: before (b) after(b) optimization. Figures 8–10 show the transient values of the current, the voltage, and the torque depending on optimization. the rotor position for the rated mode (mode № 3, 28,000 rpm, 1.5 kW). Table 2. FRM parameters (before and after the optimization).

Figure Thecurrent currentdependence dependence on position. Figure 8. The on the rotor position. Figure 8. 8. The current dependence onthe therotor rotor position.

Figure Thevoltage voltagedependence dependence on position. Figure 9. 9. The onthe therotor rotor position.

Like other single-phase motors, considered FRMs and after the optimization have Figure 9. The both voltage dependence on thebefore rotor position. a high torque ripple. However, the torque ripple of the optimized FRM is a little lower, although it was not included in the optimization criterion.

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Figures 11–13 illustrate the change in the efficiencies and losses for the three modes considered in Table 1, as well as motor losses during the optimization procedure. The change in the average efficiency (Equation (5)) is also shown. From these data, it can be concluded that, most of all, the efficiency in the least loaded mode η1 (more than 4%) was increased. The efficiency in two other modes (η2 and η3 ) was increased by 1.7% and 0.3%, respectively. The magnetic losses Pmagn (the sum of losses in steel and magnets, Figure 12) remain approximately constant (Pmangn1 is the loss in the least loaded mode; Pmangn3 is the loss in the most loaded mode). The increase in efficiency is mainly due to a decrease in loss in the stator winding Pcu (Figure 13, Pcu1 is the loss in the least loaded mode; Pcu3 is the loss in the most loaded mode). The whole optimization process requires about 110 iterations. Table 3 specifies the FRM characteristics before and after the optimization. Although the magnets’ thickness was not varied, their mass reduced during the optimization due to the reduction of the outer rotor radius. Energies 2018, 11, xwas FOR PEER REVIEW 9 of 14

Energies 2018, 11, x FOR PEER REVIEW Energies 2018, 11, x FOR PEER REVIEW Figure 10.10.The onthe therotor rotor position. Figure Thetorque torque dependence dependence on position.

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Like other single-phase motors, both considered FRMs before and after the optimization have a high torque ripple. However, the torque ripple of the optimized FRM is a little lower, although it was not included in the optimization criterion. Figures 11–13 illustrate the change in the efficiencies and losses for the three modes considered in Table 1, as well as motor losses during the optimization procedure. The change in the average efficiency (Equation (5)) is also shown. From these data, it can be concluded that, most of all, the efficiency in the least loaded mode η1 (more than 4%) was increased. The efficiency in two other modes (η2 and η3) was increased by 1.7% and 0.3%, respectively. The magnetic losses Pmagn (the sum of losses in steel and magnets, Figure 12) remain approximately constant (Pmangn1 is the loss in the least loaded mode; Pmangn3 is the loss in the most loaded mode). The increase in efficiency is mainly due to a decrease in loss in the stator winding Pcu (Figure 13, Pcu1 is the loss in the least loaded mode; Pcu3 is the loss in the most loaded mode). The whole optimization process requires about 110 iterations. Figure 11. The efficiency during the optimization procedure its average value according to (5). Table specifies thechange FRM characteristics before and after theand optimization. Although the magnets’ Figure311. 11. Theefficiency efficiency change duringthe theoptimization optimization procedure andits itsaverage average valueaccording according to(5). (5). Figure The change during procedure and value to thickness was not varied, their mass was reduced during the optimization due to the reduction of the outer rotor radius. Table 3. FRM performances (before and after the optimization).

Parameter Before After Mechanical power at the rated mode, W 1500 1500 Rated rotational speed, rpm 28,000 28,000 Outer diameter of the rotor, mm 25.8 23.4 Stator slot area, cm2 0.94 1.3 η 1, % 79.5 83.6 η 2, % 86.9 88.6 η 3, % 89.3 89.6 ηav, % 87.8 88.8 Figure magnetic losses (sum of losses in steel during optimization procedure. Figure 12. 12. TheThe magnetic losses (sum ofripple, losses in steel andand PM)PM) during the the optimization Torque % 358 335 procedure. Figure 12. The magnetic losses (sum of losses in steel and PM) during the optimization procedure. Thickness of permanent magnets (PMs), mm 1.7 1.7 Thickness of sheets of the steel lamination 0.5 0.5 (stator and rotor), mm Steel grade M330-50A M330-50A Weight of permanent magnets, g 39 35.3

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Figure 12. The magnetic losses (sum of losses in steel and PM) during the optimization procedure.

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Figure copper losses during optimization procedure. Figure 13.13. TheThe copper losses during thethe optimization procedure. Table 3. FRM performances (before and after the optimization).

In Figure 14, the total motor losses before and after the optimization are shown. In Figures 15– Parameter Before load modes.After 17, the electric losses in various parts of the motor are shown for the various In Figures 15–17, the following designations typesW of motor losses are accepted: Pcu is the 1500 loss in the Mechanical power atof thecertain rated mode, 1500 stator winding; Pse is Rated the loss in the speed, stator rpm steel from eddy currents; P28,000 sh is the loss in the stator steel rotational 28,000 due to hysteresis; Pre is the loss in the rotor steel from eddy currents; Prh is the loss in the rotor steel Outer diameter of the rotor, mm 25.8 23.4 due to hysteresis; Ppm is the loss in permanent magnets. Stator slot area, cm2

0.94

1.3

η1 , %

79.5

83.6

η2 , %

86.9

88.6

η3 , %

89.3

89.6

ηav , %

87.8

88.8

Torque ripple, %

358

335

Thickness of permanent magnets (PMs), mm

1.7

1.7

Thickness of sheets of the steel lamination (stator and rotor), mm

0.5

0.5

Steel grade

M330-50A

M330-50A

Weight of permanent magnets, g

39

35.3

Weight of stator steel, g

371

360

Weight of rotor steel, g

91

65.3

Weight of copper, g

126

175

Weight of active materials, kg

0.627

0.636

In Figure 14, the total motor losses before and after the optimization are shown. In Figures 15–17, the electric losses in various parts of the motor are shown for the various load modes. In Figures 15–17, the following designations of certain types of motor losses are accepted: Pcu is the loss in the stator winding; Pse is the loss in the stator steel from eddy currents; Psh is the loss in the stator steel due to hysteresis; Pre is the loss in the rotor steel from eddy currents; Prh is the loss in the rotor steel due to hysteresis; thePEER lossREVIEW in permanent magnets. Energies 2018,P11, FOR 11 of 14 pmxis

Figure Figure 14. 14. The The total total motor motor losses losses in in various various modes. modes.

Figure 14. The total motor losses in various modes. Figure 14. The total motor losses in various modes. Energies 2018, 11, 3334

Figure 14. The total motor losses in various modes.

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Figure 15. The motor losses at the rated load (1.5 kW, 28 krpm, “mode 1”), before and after Figure 15. The motor losses at the rated load (1.5 kW, 28 krpm, “mode 1”), before and after optimization. optimization. Figure 15. losses at the load (1.5 kW, 28 krpm, “mode 1”),“mode before and optimization. Figure 15. The Themotor motor losses at rated the rated load (1.5 kW, 28 krpm, 1”), after before and after optimization.

Figure 16. The motor losses at the partial load (0.562 kW, 21 krpm, “mode 2”), before and after Figure 16. 16. The Themotor motor losses partial (0.562 21 krpm, “mode 2”), before and Figure losses at at thethe partial loadload (0.562 kW,kW, 21 krpm, “mode 2”), before and after optimization. after optimization. optimization. Figure 16. The motor losses at the partial load (0.562 kW, 21 krpm, “mode 2”), before and after optimization.

17. The Themotor motor losses partial (0.187 14 krpm, “mode 3”), before and Figure 17. losses at at thethe partial loadload (0.187 kW,kW, 14 krpm, “mode 3”), before and after Figure 17. The motor losses at the partial load (0.187 kW, 14 krpm, “mode 3”), before and after after optimization. optimization. optimization. Figure 17. The motor losses at the partial load (0.187 kW, 14 krpm, “mode 3”), before and after The reduction reductionofofthe thetotal total losses to optimization significant at underload. The losses duedue to optimization was was most most significant at underload. Thus, optimization.

The reduction ofwere the total losses due to optimization was 1, most significant at at underload. Thus, Thus, thelosses total losses decreased W (24%) at mode by W at (15%) mode the total were decreased by 11.8byW11.8 (24%) at mode 1, by 12.7 W12.7 (15%) mode 2, and 2, byand 5.1 by W the total losses were 3. decreased bycontribution 11.8 W (24%)toatthe mode 1, by 12.7 W (15%) at mode 2, and byby 5.1the W 5.1 W at3.mode The main reduction of losses the total losses provided (3%) at(3%) mode Theof main contribution to the of the total is provided by the reduction The reduction the total losses due to reduction optimization was most significant atisunderload. Thus, (3%) at mode 3. The main losses, contribution to the reduction of the totalthe losses is provided bystator the reduction reduction of the winding which isby obtained by increasing and slot5.1 area. of the winding losses, which is by obtained increasing the1,copper and mass stator slot the total losses were decreased 11.8 W (24%) at mode by 12.7mass Wcopper (15%) at mode 2,area. and by W of theThe winding losses, which is obtained by increasing the copper mass and stator slot area. core losses exhibited insignificant changes. However, they also should be taken into account The core3. losses exhibited insignificant However, they also should be taken into account (3%) at mode The main contribution to thechanges. reduction of the total losses is provided by the reduction The core losses exhibited insignificant changes. However, they alsocore should be and taken account when the Hence, to decrease further losses tointo increase the when calculating theefficiency. efficiency. Hence,the the onlyway way to decrease further core losses to increase of the calculating winding losses, which is obtained byonly increasing the copper mass and stator slotand area. when calculating the efficiency. Hence, the only way to decrease further core losses and to increase efficiency is by using thinner steel sheets with lower specific losses, and, in particular, eddy current The core losses exhibited insignificant changes. However, they also should be taken into account losses.calculating The core losses are significant in the and they arefurther very small underload the when the efficiency. Hence, therated only mode way to decrease core at losses and towhile increase winding losses prevail in underload. Therefore, a further increase in the rated efficiency is possible with the use of thinner steel in the manufacture of magnetic cores. The losses in magnets are very small. So, the number of parts in which magnets are divided can be slightly increased without significantly decreasing the efficiency if it reduces the FRM production cost. Although the windage losses constitute a significant part in the total losses, especially when a high-speed motor has a toothed rotor, they were not taken into account. However, during the optimization, the rotor slot became smaller. Therefore, the reduction of the windage losses is expected.

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5. Conclusions This paper describes the design of a high-speed single-phase flux reversal motor for vacuum cleaners. This machine has a simple and reliable rotor, which is a significant advantage for high-speed applications. In addition, it requires a simple and cheap inverter. An FRM design in which the inner stator surface is entirely used, and which allows it to decrease in volume and increase in efficiency, is described. The optimization of the single-phase FRM by the Nelder–Mead method using the FEM model is provided. The optimization criterion was selected so as to maximize the efficiency in a wide range of powers at the fan load (quadric dependence of torque on speed). Various modes of that load profile were supposed to be equally used. The reduction of the total losses due to optimization was most significant at underload. Thus, the total losses were decreased by 11.8 W (24%) at mode 1 (the least loaded mode), by 12.7 W (15%) at mode 2, and by 5.1 W (3%) at mode 3 (the rated mode). The main contribution to the reduction of the total losses is provided by the reduction of the winding losses, which is obtained by increasing the copper mass and stator slot area. Losses in steel in the rated mode are significant. Therefore, a further increase in the rated efficiency is possible with the use of thinner steel in the manufacture of magnetic cores. In future works, a theoretical and experimental comparison of the single-phase FRM to other types of high-speed single-phase motors [4–6] will be conducted. It is also planned to test the prototype of the single-phase FRM as a part of a vacuum cleaner or air blower. Author Contributions: All the authors contributed substantially to the paper. Modeling and optimizing the FRM was done by V.D. V.P. provided the conceptual approach and provided comments at all the stages of the work. V.K. prepared the graphs, pictures, and tables and provided comments at all the stages of the paper. S.S. revised the manuscript. Funding: The research was conducted on theme no. NO 8.9549.2017/8.9. within the frame of the government task of the Ministry of Education and Science of the Russian Federation in R&D. Conflicts of Interest: The authors declare no conflict of interest.

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